
The condition (A4) states that the maximum is attained at a single point. It can be deduced from the Bulinskaya lemma (Proposition 6.11 in [1]}) since for \(s\ne t\) , \((X(s),X(t),X^{\prime }(s),X^{\prime }(t))\)  has a non-degenerate distribution.

The condition (A5) states that there is almost surely no point \(t\in S\)  such that \(X^{\prime }(t)=0\)  and \(\det \left(X^{\prime \prime }(t)\right)=0\) . It can be deduced from Proposition 6.5 in [1]} applied to the process, \(X^{\prime }(t)\) , which is \(\mathcal {C}^2\) .

